Abstract
What factors affect the value of a security? That is a common, and important, question asked by investors across all asset classes. Investors of fixed-income securities are no exception. Security holders take risk. They must be, at least in expectation, compensated for taking this risk. Factors that affect the value of a security, therefore, are those factors that generate risk or, in other words, return or loss.
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George Pólya
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Notes
- 1.
In reality, market practice is almost always a bit more complicated than it appears in (2.1). There are day-count conventions that describe each of the individual (t i − t)’s, compounding frequencies impacting the yield and the coupon, and settlement dates. These elements are market convention. For the purposes of this discussion, however, we will skip over many of these details unless, of course, they become important.
- 2.
We employ the terms discounted cash flow and present value of a cash flow interchangeably.
- 3.
This is formally termed a central finite-difference approximation. See Press et al. [1] for much more information on the numerical computation of derivatives.
- 4.
Caution should nevertheless be exercised as there is not, to the author’s knowledge, a clear consensus in the finance universe on the definition of effective duration.
- 5.
In general, given that the numerical computation requires three full function valuations, it will only be employed in the absence of an analytical solution.
- 6.
It is often the case, for example, that clean prices are returned from various software functions and, as a consequence, a bit of caution is advised.
- 7.
Note also that an increase of 100 basis points generates a $4.67 decrease in the price, while a 50 basis-point decrease leads to a $4.93 basis-point rise in the price. There are two points that should be taken from this fact. First, a 100 basis-point movement in yields leads to relatively large price changes. Second, the price movement is not symmetric for an equivalent upward and downward movement in bond yields. This is due to the fact that the relationship between bond prices and yields is not linear. More on this point will be discussed in the next section.
- 8.
The product of the security’s yield and the time interval approximates the return associated with the movement of time.
- 9.
This computation is based on the fact that modified duration—essentially a derivative—is a linear quantity and, as such, can be averaged.
- 10.
This is because all terms where i ≠k are, by definition, zero. That is, they do not contribute to the derivative.
- 11.
In words, the product of the key-rate duration and an expected or realized change in the kth yield approximates the approximate percentage change in the value of one’s fixed-income instrument.
- 12.
For more information on this technique, see Chap. 5.
- 13.
For a portfolio with numerous bonds across the curve, the effect is likely to be relatively small. Nevertheless, for computations that require a high degree of accuracy, such as performance attribution, this is likely to remain an unacceptable source of error.
- 14.
This latter situation is also termed credit migration.
- 15.
One could, for example, use categories such as sovereign, agency, supranational, or corporate. It would also be possible to have subcategories with each of these groups or even, in the limit, focus on individual names.
- 16.
To be precise, the spread may also include additional yield demanded by lenders to compensate for lower liquidity of the credit bond relative to the underlying government curve. Decomposition of the credit and liquidity aspects of the credit spread is not trivial and we will, in our development ignore such effects. It is nonetheless important to be aware that part of the credit spread may be attributable to liquidity and, in some cases, its contribution can be important.
- 17.
The explicit introduction of the credit spread into the bond price equation permits us to include the credit risk factor into our mathematical framework.
- 18.
A possible exception would be a dual-currency bond, where the coupon and notional amounts are denominated in different currencies. In this case, the security would need to be split into two separate instruments: an annuity for the coupon stream and a zero-coupon bond for the notional value. Each would then require a separate currency definition.
Reference
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. (Cambridge University Press, Cambridge, 1992)
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Bolder, D.J. (2015). Computing Exposures. In: Fixed-Income Portfolio Analytics. Springer, Cham. https://doi.org/10.1007/978-3-319-12667-8_2
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DOI: https://doi.org/10.1007/978-3-319-12667-8_2
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